I am attempting Silverman's AEC exercise I.1.7 part (c).
Instead of using intrinsic definitions or results, I am trying working with the definitions stated in the book, i.e. the two definitions in Section 1.3 about rational maps and regular points. As the same as in the link, I found out that $[X,Y,Z] \mapsto [Y,X]$ is the formula of $\psi$, and I also predict that $\psi$ is not a morphism. More explicitly, I will show that $\psi$ is not regular at $[0,0,1]$. Suppose so, then $\psi$ can be written as a pair $[\psi_0,\psi_1]$ where $\psi_0,\psi_1$ are homogeneous three-variable ($x,y,z$) polynomials of the same degree, such that any of them has non-zero value at $[0,0,1]$ and $X\psi_1-Y\psi_0$ lies in $I(V)$. From this point I have two approaches, which are in fact not really different:
Plugging in some triples $(X,Y,Z)$ and see if they can lead to any contradictions. I tried $(0,0,1)$ but it failed since we don't have any "$Z$" here.
Note that $\phi$ is a bijection, so a point in $V$ must be of the form $[S^2T,S^3,T^3]$ where $S,T$ are parameters not equal to zero at the same time. From this I change $X\psi_1-Y\psi_0$ to a polynomial of two variable $S,T$ which is $$S^2T\psi_1(S^2T,S^3,T^3)-S^3\psi_0(S^2T,S^3,T^3) \in I(V).$$ Then dividing two sides by $S^2$ and plugging in $S=0,T=1$ gives me $\psi_1([0,0,1])=0$ and $\psi_0([0,0,1])=\lambda\neq 0$. From here I don't know what to do next since I have no idea how to bound the degree of $\psi_0$ and $\psi_1$.
Any help is really appreciated.